📚 Exercise 17A.1: Differentiation of Trigonometric Functions | 练习 17A.1：三角函数的微分
In IB Mathematics, Exercise 17A.1 typically focuses on differentiating trigonometric functions – a fundamental skill in calculus. Understanding how to find derivatives of sin x, cos x, tan x, and their compositions using the chain, product, and quotient rules is essential for success in both Analysis & Approaches and Applications & Interpretation courses.
在 IB 数学中，练习 17A.1 通常聚焦于三角函数的微分——微积分中的基本技能。理解如何求 sin x、cos x、tan x 的导数，并运用链式法则、乘积法则和商法则处理复合函数，对于在分析与方法和应用与解释课程中取得成功至关重要。
1. Review of Basic Differentiation Rules | 基本微分法则回顾
tandard notation. The derivative of xⁿ is nxⁿ⁻¹. Constants factor out: d/dx [c f(x)] = c f' y of these fundamentals ensures smooth handling of more complex trigonometric expressions.
在深入三角函数导数之前，回顾幂法则、常数倍法则和标准符号。xⁿ 的导数为 nxⁿ⁻¹。常数可以提出：d/dx [c f(x)] = c f'(x)。和法则允许逐项求导，常数的导数为零。掌握这些基础能确保顺利处理更复杂的三角表达式。
d/dx (xⁿ) = n xⁿ⁻¹
2. Derivative of Sine Function | 正弦函数的导数
xpanding sin(x+h) = sin x cos h + cos x sin h, factoring, and applying the special limits.
sin x 的导数是 cos x。该结果依赖于导数的极限定义和正弦和角公式。在弧度制下，基本极限 lim(h→0) (sin h)/h = 1 起着关键作用。证明通过展开 sin(x+h) = sin x cos h + cos x sin h、提取公因式并应用特殊极限得出。
d/dx (sin x) = cos x
3. Derivative of Cosine Function | 余弦函数的导数
The derivative of cos x is –sin x. Note the negative sign – urce of errors. One way to derive this is by using the cofunction identity cos x = sin(π/2 – x). Applying the chain rule gives d/dx [sin(π/2 – x)] = cos(π/2 – x) · (–1) = –sin x by another cofunction identity.
cos x 的导数是 –sin x。注意负号——这是常见的错误来源。推导的一种方法是利用余函数恒等式 cos x = sin(π/2 – x)。应用链式法则可得 d/dx [sin(π/2 – x)] = cos(π/2 – x) · (–1) = –sin x，再通过另一个余函数恒等式化简。
d/dx (cos x) = –sin x
4. Derivative of Tangent Function | 正切函数的导数
otient rule on tan x = sin x / cos x. Numerator derivative: cos x, denominator derivative: –sin x. Plugging into (u'v – uv')/v² yields (cos² x + sin² x)/cos² x = 1/cos² x = sec² x.
tan x 的导数是 sec² x，等价于 1/cos² x。最简洁的推导是对 tan x = sin x / cos x 使用商法则。分子导数为 cos x，分母导数为 –sin x。代入 (u'v – uv')/v² 得到 (cos² x + sin² x)/cos² x = 1/cos² x = sec² x。
d/dx (tan x) = sec² x
5. The Chain Rule with Trigonometric Functions | 三角函数的链式法则
functions like sin(2x) or cos(x² + 1), the chain rule is indispensable: d/dx [f(g(x))] = f'(g(x)) · g'(x). For a general sine composite, d/dx [sin(u)] = cos(u) · u' (3x) = cos(3x) · 3 = 3 cos(3x). Always identify the inner function before differentiating.
当对 sin(2x) 或 cos(x² + 1) 等复合函数求导时，链式法则是不可或缺的：d/dx [f(g(x))] = f'(g(x)) · g'(x)。对于一般的正弦复合，d/dx [sin(u)] = cos(u) · u'。例如，d/dx sin(3x) = cos(3x) · 3 = 3 cos(3x)。在求导前一定要先识别出内函数。
d/dx [sin(u)] = cos(u) · u'
6. Product Rule Involving Trigonometric Expressions | 涉及三角表达式的乘积法则
The product rule states d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v' erentiate x² sin x. Here u = x², v = sin x, giving 2x sin x + x² cos x. Another example: eˣ cos x yields eˣ cos x – eˣ sin x.
乘积法则指出 d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)。当两个因子都包含三角函数时，需仔细求导以避免符号错误。例如，求 x² sin x 的导数。此处 u = x², v = sin x，得到 2x sin x + x² cos x。另一个例子：eˣ cos x 的导数为 eˣ cos x – eˣ sin x。
d/dx [x² sin x] = 2x sin x + x² cos x
7. Quotient Rule with Trigonometric Ratios | 含有三角比的商法则
The quotient rule is d/dx [u(x)/v(x)] = (u'v – uv' lly useful for differentiating tan x (already seen) as well as cot x, sec x, and csc x. It's also applied to rational expressions like sin x / (x+1). In that case, u' = cos x, v' = 1, leading to (cos x (x+1) – sin x) / (x+1)².
商法则是 d/dx [u(x)/v(x)] = (u'v – uv') / v²。这对于求导 tan x（已见）以及 cot x、sec x 和 csc x 尤其有用。它也应用于有理表达式，如 sin x / (x+1)。此时 u' = cos x, v' = 1，得到 (cos x (x+1) – sin x) / (x+1)²。
A handy summary of derivatives for all six basic trigonometric functions is given below.
以下是六个基本三角函数导数的方便总结。
| Function f(x) | Derivative f'(x) |
|---|---|
| sin x | cos x |
| cos x | –sin x |
| tan x | sec² x |
| cot x | –csc² x |
| sec x | sec x tan x |
| csc x | –csc x cot x |
8. Implicit Differentiation and Trigonometric Functions | 隐函数微分与三角函数
se implicit differentiation. Differentiate both sides with respect to x, treating y as a function of x. For sin(x+y), the derivative is cos(x+y)·(1 + dy/dx). The right side gives y + x dy/dx. Then solve for dy/dx.
如果方程在三角函数内混合了 x 和 y，例如 sin(x + y) = xy，我们需要使用隐函数微分。对等式两边关于 x 求导，将 y 视为 x 的函数。对于 sin(x+y)，导数为 cos(x+y)·(1 + dy/dx)。右边得到 y + x dy/dx。然后解出 dy/dx。
d/dx [sin(x+y)] = cos(x+y)·(1 + dy/dx)
9. Higher-Order Derivatives of Trigonometric Functions | 三角函数的高阶导数
The second derivative f'' trigonometric functions, repeated differentiation cycles through four stages. For sin x: f'(x)=cos x, f''(x)=–sin x, f'''(x)=– ⁽⁴⁾(x)=sin x. This cyclic behaviour is useful in differential equations and Taylor series.
二阶导数 f''(x) 衡量斜率的变化率。对于三角函数，重复求导会经历四个阶段的循环。对于 sin x：f'(x)=cos x, f''(x)=–sin x, f'''(x)=–cos x, f⁽⁴⁾(x)=sin x。这种循环性质在微分方程和泰勒级数中非常有用。
sin x → cos x → –sin x → –cos x → sin x
10. Worked Examples from Exercise 17A.1 | 练习 17A.1 典型例题详解
Let's work through typical problems students encounter. (a) Differentiate f(x) = 3 sin(2x) – x. Apply the chain rule to the first term: 3·cos(2x)·2 = 6 cos(2x). The second term gives –4·(–sin x) = 4 sin x. Hence f' n dy/dx = sec²(x²)·2x. (c) Find dy/dx for y = x³ sin(5x). Using product and chain rules: y' = 3x² sin(5x) + x³·cos(5x)·5 = 3x² sin(5x) + 5x³ cos(5x).
我们来演练学生遇到的典型题目。(a) 求 f(x) = 3 sin(2x) – 4 cos x 的导数。对第一项应用链式法则：3·cos(2x)·2 = 6 cos(2x)。第二项得 –4·(–sin x) = 4 sin x。因此 f'(x) = 6 cos(2x) + 4 sin x。(b) 若 y = tan(x²)，则 dy/dx = sec²(x²)·2x。(c) 求 y = x³ sin(5x) 的 dy/dx。使用乘积法则和链式法则：y' = 3x² sin(5x) + x³·cos(5x)·5 = 3x² sin(5x) + 5x³ cos(5x)。
y = x³ sin(5x) → y' = 3x² sin(5x) + 5x³ cos(5x)
11. Common Mistakes and Tips | 常见错误与提示
Here are pitfalls to avoid: forgetting the chain rule's inner derivative (e.g., differentiating sin(2x) as cos(2x) only), using degrees instead of radians (derivative formulas assume radian measure), dropping the negative sign when differentiating cos x, and misapplying the quotient rule order. Always double-check the sign, use parentheses to separate terms, and practise identifying the inside function.
以下是需要避免的陷阱：忘记链式法则的内函数导数（如将 sin(2x) 仅求导为 cos(2x)）、使用角度制而非弧度制（导数公式基于弧度）、余弦求导时丢失负号、以及商法则分子减法的顺序错误。务必反复检查符号，使用括号分隔各项，并练习识别内函数。
12. Consolidation Practice Approach | 巩固练习方法
After mastering the basic derivatives, try to differentiate a mixture of functions without hesitating. Write down the rule you are applying at each step. Start with simple powers and trig, then add exponentials and logarithms as they appear in later exercises. Remember that all derivatives can be checked by integration or technology to build confidence.
在掌握基本导数后，试着毫不犹豫地对混合函数求导。在每一步写下你应用的法则。从简单的幂函数和三角函数开始，然后在后续练习中引入指数和对数函数。记住，所有导数都可以用积分或技术工具进行验证，以建立信心。
Practice makes the differentiation of trigonometric functions second nature.
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